11.08 Fitting lines to bivariate data
Scientists conducted a study where each person was asked to read a paragraph and then recount as much information as they can remember. They found that the longer the paragraph, the less information each person could retain.
If the length of the paragraph were plotted (on the horizontal axis) against the amount of information retained (on the vertical axis), would the graph show a positive or negative correlation?
Consider the table of values that show four excerpts from a database comparing the income per capita of a country and the child mortality rate of the country.
If a scatterplot was created from the entire database, would you expect the relationship to be a strongly positive, strongly negative or have no relationship?
| Income per capita | Child mortality rate |
|---|---|
| 3041 | 80 |
| 10\,841 | 20 |
| 12\,997 | 33 |
| 32\,262 | 8 |
The following table shows the number of traffic accidents associated with a sample of drivers of different age groups:
| Age | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | 65 |
|---|---|---|---|---|---|---|---|---|---|---|
| Accidents | 41 | 44 | 39 | 34 | 30 | 25 | 22 | 18 | 19 | 17 |
Construct a scatter plot for this data.
Describe the correlation between a person's age and the number of accidents.
Does this data set contain any outliers?
The following table shows the average IQ of a random group of people against their height:
| \text{Height (cm)} | 140 | 145 | 150 | 155 | 160 | 165 | 170 | 175 | 180 | 185 |
|---|---|---|---|---|---|---|---|---|---|---|
| \text{IQ} | 103 | 95 | 98 | 111 | 85 | 89 | 108 | 145 | 110 | 93 |
Construct a scatterplot using the data from the table.
Is IQ and height negatively correlated, postively correlated or not correlated?
How tall is the person who appears to be an outlier?
A researcher is studying the relationship between the number of passers-by present in a situation and the time taken, in seconds, until a stranger in an emergency receives help from a passer-by. The data is recorded in the table below:
| \text{Number of passers-by (n)} | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| \text{Time until help is offered (t)} | 8 | 19 | 26 | 37 | 51 | 65 |
Construct a scatterplot using the data from the table.
As more passers-by are present what happens to the time taken until help is offered?
Describe the correlation between the number of passers-by and the time until assistance is offered.
Does the scatterplot contain any outliers?
The following table shows the marks of 12 students in Maths and Sports:
Construct a scatter plot for the students' marks in Maths vs their marks in Sports.
Describe the correlation between students' marks in Maths and marks in Sports.
Which student's scores appear to represent an outlier?
| Student | Maths | Sports |
|---|---|---|
| 1 | 63 | 44 |
| 2 | 92 | 74 |
| 3 | 60 | 52 |
| 4 | 79 | 70 |
| 5 | 88 | 67 |
| 6 | 81 | 60 |
| 7 | 61 | 73 |
| 8 | 91 | 86 |
| 9 | 72 | 84 |
| 10 | 42 | 93 |
| 11 | 66 | 57 |
| 12 | 92 | 92 |
The following table shows the marks of 12 students in English and French:
Construct a scatter plot for English mark vs French mark.
Describe the correlation between students' English and French marks.
Which student's scores appear to represent an outlier?
| Student | English | French |
|---|---|---|
| 1 | 85 | 89 |
| 2 | 71 | 71 |
| 3 | 57 | 56 |
| 4 | 60 | 62 |
| 5 | 79 | 86 |
| 6 | 76 | 76 |
| 7 | 71 | 77 |
| 8 | 91 | 86 |
| 9 | 50 | 90 |
| 10 | 49 | 47 |
| 11 | 66 | 67 |
| 12 | 92 | 92 |
The table lists the time taken to sprint 400 \text{ m} by runners who all ran in different temperatures as part of a study:
Construct a scatter plot to represent the data in the table.
How many runners were tested in the study?
Describe the correlation between temperature and sprint time for the data.
Which data point represents an outlier?
| \text{Temperature } \\ \left(\degree \text{C} \right) | \text{Time (sec)} |
|---|---|
| 5 | 60 |
| 2 | 67 |
| 10 | 48 |
| 8 | 69 |
| 1 | 65 |
| 7 | 49 |
| 6 | 57 |
| 4 | 53 |
| 3 | 59 |
| 9 | 52 |
To determine whether the presence of sharks in a coastal region is influenced by cage diving, the number of nearby cage diving operations and the number of nearby shark sightings was recorded each month over several months. The results are shown in the table below:
| Cage diving operations | 4 | 1 | 6 | 5 | 2 | 3 |
|---|---|---|---|---|---|---|
| Shark sightings | 2 | 6 | 3 | 7 | 3 | 8 |
Construct a scatterplot using the data from the table.
Describe the correlation between the number of shark sightings and the number of cage diving operations.
According to the data, is it possible to determine whether cage diving operations encourage more sharks to come near the shoreline?
The market price of bananas varies throughout the year. Each month, a consumer group compared the average quantity of bananas supplied by each producer to the average market price (per unit).
Construct a scatterplot using the data from the table.
Describe the correlation between the supply quantity and the market price of bananas.
According to this data, when will a supplier of bananas receive a higher price per banana?
| \text{Supply (kg)} | \text{Price (dollars) } |
|---|---|
| 550 | 16.25 |
| 600 | 15.75 |
| 650 | 15.75 |
| 700 | 14.75 |
| 750 | 14.50 |
| 800 | 14.00 |
| 850 | 13.75 |
| 900 | 12.75 |
| 950 | 12.50 |
| 1000 | 12.00 |
The scatter plot shows the relationship between air and sea temperature:
Describe the relationship between air and sea temperature.
Describe the correlation between air and sea temperature.
The following scatter plot shows the relationship between sea temperatures and the amount of healthy coral:
Describe the correlation between sea temperature the amount of healthy coral.
Which variable is the response variable?
Which variable is the explanatory variable?
Draw an approximate line of best fit by hand for each of the the scatter plots below:
The following scatter plot graphs data for the number of people in a room and the room temperature collected by a researcher:
Draw an approximate line of best fit for this scatter plot.
The following scatter plot graphs data for the number of copies of a particular book sold at various prices:
Draw an approximate line of best fit for this scatter plot.
For each equation for the line of best fit:
State the gradient of the line.
Describe the correlation of the data set as positive or negative based from the gradient.
Describe the change in y as x increases by 1 unit.
State the value of the y-intercept.
The average monthly temperature and the average wind speed, in knots, in a particular location was plotted over several months. The graph shows the points for each month’s data and their line of best fit:
Use the line of best fit to approximate the wind speed on a day when the temperature is 5\degree \text{C}.
The following scatter plot shows the data for two variables, x and y:
Sketch the line of best fit for this data.
Use your line of best fit to estimate the value of y when:
x = 4.5
x = 9
The following scatter plot graphs data for the number of ice blocks sold at a shop on days with different temperatures.
Sketch the line of best fit for this data.
Use your answer line of best fit to estimate the number of ice blocks that will be sold on a:
31 \degree \text{C} day
42 \degree \text{C} day
Does the number of ice blocks sold increase or decrease as the temperature increases?